Conditional κ = 1.002661106042177·10⁻³: our #394 padding fusion on #398's R781 signed-prefix E8 complex program (+0.0166 % over #398)

[chafreaky]

e8-clean-r781: our #394 padding fusion on #398's signed target-prefix R781 E8 complex program; conditional κ = 1.002661106042177e-3

κ = 1002661106042177/10¹⁸ ≈ 1.002661106042177 × 10⁻³, complex-bound.

• vs Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 (Dugongue, 1.002494659629864e-3): +1.66446412313·10⁻⁷ (+0.0166 %).
• Complex side: b rises from Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's 31359395740192012461/3.125·10²² ≈ 1.003500664e-3 to 1003667444352030197427/10²⁴ ≈ 1.003667444e-3 (+1.66781·10⁻⁷).
• Bit side: our Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382 word, c ≈ 1.006907e-3 > b. Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's bit c gives the same κ.

What is new

#398's own bracket rules fusion out. #398 compiles its R781 word with #352's frozen old response K plus the full non-y cleanup at FULL. Every one of its 781 helpers is the target or source of a non-y gate (checked), so each is touched after its final climb, and no padding can be fused there.

#390's dirty-preconditioned bracket allows it. We compile the same word with #390's bracket: precancellation y −= B·A⁻¹·h, prewhitening, and FULL repair of non-sink helpers only. Our compiler reproduces #390's prewhite.json on #390's own word.

• On Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's word it is cost-neutral: the emitted paid local histogram equals the word's.
• From our emission, Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's b and κ are reproduced exactly.
• Free zero-row sinks then need no cleanup, and our Conditional κ = 1.000385913854264·10⁻³: bank padding fused into three new free sink helpers on #390's E8 complex program (+0.0166 % over #390) #394 fusion applies.

The free-sink search over all 773 lifting helpers of #398's word finds exactly 301, 324 and 387 (final lift rank 2; frames 731/744/759; cuts 1974/2005/2040). For 301 the gates are h276 += h1019/2; h301 −= h276/2; h276 −= h1019/2.

Under #398's whole-bank rule (bank, slot = divmod(rep·781 + r, 8), T = 8), the 24 fused occurrences per stage fall in 24 distinct banks. Every slot route fixes coordinate 72. Paddings per stage drop from 781 to 757, and the profile delta is {1: −216, 2: −216, 3: +216}. Live 14,709, deficit 2,496 and max child 48 are unchanged.

Checks (bash research/e8-clean-r781/verify.sh [--quick] /fresh/dir; Python 3.11+, stdlib only, no C++)

• Pins: git-blob pins on Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's signed25.json.gz and VALIDATION.json, E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's prefix18, sink-audit and prewhite, Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378's word, Sussman's gx, and Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382's word.
• Bracket regression: our compiler reproduces E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's prewhite.json gate for gate.
• Words: gx.check1 (E0–E7, exact scalar), the gxcore mirror, the flipped-sign control and the source-span rule pass on Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's word and on ours.
  • Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's x → (x, y) operator equals Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378's.
  • Ours keeps Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's paid histogram, x/y operator rows and retained totals, and its sink rows are exactly zero.
  • Full mode only: the free-sink search regenerates exactly our gates.
• Local program: Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's word and ours are replayed in both orientations on all 1,021 live columns (private work erased, common exact labels, no event on a fused sink after its padded climb). Five mutation controls are rejected.
• Banks: fused occurrences are in distinct banks, and coordinate 72 is fixed.
• Price, with Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 as the control: Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's 24-digit coarse root (both PR Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315 engines, 10⁻¹⁶ fallback) and κ are reproduced exactly. The κ uses the Conditional kappa 0.001000220224856110: six fixed levels on #390 #392/Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 six-level bootstrap with PR Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315's outer assembly (47 strict constraints, next grid point rejected). Ours is priced the same way.

Local run on commit b974e48, from a fresh clone: --quick and full both PASS, and the tree is unchanged afterwards.

Conditions

• Re-opened (PROOF.md):
  • Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's literal K/cleanup bracket is replaced by E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's bracket. It is checked on all columns in both orientations. Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's Gaussian-phase, global-template and finite-invoice audits were run for its own scalar schedule and are not re-run for ours, so the invoice must be re-counted for the changed scalar work. E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390's invoice for this bracket on R783 stayed below the reserve;
  • κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's padding as a separate sweep, for the 24 fused banks per stage.
• Inherited, not re-run:
  • Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's R781 reconstruction (Conditional κ=0.0011208649522582786: split centres with complete signed admission #376) and its Clifford edge audits;
  • Conditional kappa 0.001000220224856110: six fixed levels on #390 #392's bootstrap argument;
  • κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377's and κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352's interfaces;
  • Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382's bit claims.

This is a conditional finite construction, not a formal verification.

Credits

• Dugongue: Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398 (the signed target-prefix R781 construction, banks and invoice) and E8 prefix restoration and heterogeneous helper banks: conditional kappa = 1.000220222889528e-3 #390 (the target-prefix mechanism and the dirty-preconditioned bracket).
• applecrusher (Conditional κ=0.0011208649522582786: split centres with complete signed admission #376): the R781 helper substitutions.
• rohanarun: Conditional kappa 0.001000220224856110: six fixed levels on #390 #392/Conditional kappa 0.000998634658090835: finite bootstrap and exact coarse precision #387 (bootstrap pricing) and Conditional kappa 0.001001256225581264: signed prefixes and padding fusion #395 (signed prefixes).
• jacklightChen (κ = 9.98243154231047e-4: nine-stage complex E8 (≈2^-9.9683; complex +14.036%, bit side pending) #377): the nine-stage component, banks and padding.
• κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352: the local invocation, compiler and emitter.
• Jacob Sussman: the E8 unit and gx/gxcore.
• DaysSky (Complex supplier b = 9.32783e-4: E8 in a seven-stage bridged word (+6.45 % over #352; no new κ, raises the complex cap) #363/κ = 9.3191395651209e-4: #370's E8 bit word and Sussman's E8 complex unit in the seven-stage bridged word (+6.45 % over #370) #373/κ = 9.5424356568133e-4: seven-stage E8 suppliers with the complex idle climbs at halving degree two (+2.40 % over #373) #375/κ = 9.95656966441037e-4: #370's E8 bit word in the nine-stage bridged word with #377's nine-stage complex E8 (above 2^-10; +2.94 % over #380) #381) and utcorvusvolat-dotcom (κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370): the bit lineage.
• DreamingOfClouds (PR Conditional κ = 7.64230861320245e-4: the paired-cube bit word at cube size p = 10 (h = 20, m = 100), with a new centre-sharing complex supplier (+1.21% over #314) #315): the engines and outer assembly.
• Ours: Conditional κ = 9.5464544449649·10⁻⁴: Sussman's E8 complex unit re-timed by far piece relocation, in #375's unsplit B₃ (+0.042 % over #375) #378, Conditional κ = 9.98243154231047·10⁻⁴: our retimed + kernel E8 bit word in B₃ (unsplit) with #377's nine-stage complex component (+0.26 % over #381) #382, Conditional κ = 1.000385913854264·10⁻³: bank padding fused into three new free sink helpers on #390's E8 complex program (+0.0166 % over #390) #394, and here the bracket transfer to Title: Signed target-prefix cancellation on R781: conditional kappa = 0.001002494659629864 #398's word and the checks.

Prepared by Chafik Boukhalfa (chafreaky) with substantial Anthropic Claude assistance.

Commit b974e481efd2f771cda2dcdd71c91f133793558f on branch research/e8-clean-r781, on top of our #394 (be2a06b).

Repository CI. This branch includes PR #318's unchanged three-file joint-dual source-pin repair (certificates/joint-dual-kappa.json, certificates/joint-dual-validation.json, research/joint-dual/SOURCE.json), as #366 and #368 do, so the repository-wide verify (joint) job passes. No mathematics is changed by it.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Padding fusion on the R781 signed-prefix word.

Review and limits: Exact local checker and price contribution, with native phase/owner/invoice qualifications retained. Parallel upstream improvements remain credited.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 3e04107b9381d957a7da7e632ddb662b518429c8. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.